The distance function from a real algebraic variety

Giorgio Ottaviani, Luca Sodomaco*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

4 Citations (Scopus)
10 Downloads (Pure)


For any (real) algebraic variety X in a Euclidean space V endowed with a nondegenerate quadratic form q, we introduce a polynomial EDpolyX,u(t2) which, for any u∈V, has among its roots the distance from u to X. The degree of EDpolyX,u is the Euclidean Distance degree of X. We prove a duality property when X is a projective variety, namely EDpolyX,u(t2)=EDpolyX,u(q(u)−t2) where X is the dual variety of X. When X is transversal to the isotropic quadric Q, we prove that the ED polynomial of X is monic and the zero locus of its lower term is X∪(X∩Q).

Original languageEnglish
Article number101927
Publication statusPublished - Oct 2020
MoE publication typeA1 Journal article-refereed


  • Euclidean distance
  • Euclidean Distance degree
  • Euclidean Distance polynomial
  • Isotropic quadric
  • Polar degrees
  • Real algebraic variety


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